Second Quantization - Quasiparticles

[LarryC]

Homework Statement

Find a general formula to the first order of the inter-particle interaction for the energy of an N+1 particle system of spin-1/2 fermions with one particle of momentum p outside of an N-particle Fermi sea.

Relevant Equations

Second quantization of fermions

(Simplified version of Baym, Chapter 19, Problem 2)
Calculate, to first order in the inter-particle interaction V(r-r'), the energy of an N+1 particle system of spin-1/2 fermions with on particle of momentum p outside an N-particle Fermi sea (quasiparticle state). The answer should be expressed as the difference of the energy of the specified state and the energy of the N-particle ground state. (Probably the easiest way is to use creation and annihilation operators to express the difference as the matrix element of a difference of operators - a difference that can be simplified using anti-commutation relations.)

 

[Charles Link]

I can probably help you with this. I have the book by Baym, and tried to work through a calculation in 1979 involving equation 20-41 p.449. After getting stuck for about 2 weeks or more, I finally determined Baym has a typo in equation 19-41 on page 420. (I have the 6th printing). That equation should have $\frac{1}{\sqrt{n}}$ instead of $\frac{1}{\sqrt{n!}}$.
So much trouble caused by an exclamation mark!=factorial symbol. Once I figured that out, my calculation with equation 20-41 worked. Starting with 19-43 and the corrected 19-41, and 19-42, looking at equation 19-50, I was able to get equation 20-41 to work.
I still have that original paper. Let me take a photo of both sides of the paper, and upload it...

 

[Charles Link]

 

[Charles Link]

In the photos above, the lower image is the front side of the paper, and the upper image is the back side. The images aren't real clear, but perhaps you can follow the calculation.
To see what I did here, I took the expression at the top of the front side of the paper, the left side of equation (20-41), which leaves off the $|\Phi>$ etc., and I inserted the identity operator for $n$ particles, (equation 19-47), and then the summed over $n$ from $0$ to $+\infty$.
The rest is a lot of algebra using 19-41, 19-42, 19-43, and 19-50 to get 20-41 to work.